Usually, covers albums feature the most popular songs. On Carrie Rodriguez's third album "Love and Circumstance," there are a few cuts that were embraced by the mainstream, plus a few that made a splash generations ago. On it, Rodriguez, 31, offers up a nice, spare version of the Hank Williams classic "I'm So Lonesome I Could Cry" and delivers a sturdy version of Merle Haggard's "I Started Loving You Again."

Yet the fiddle-playing singer and songwriter shines brightest when she takes on the obscure. Her versions of Little Village's "Big Love" and Gillian Welch and David Rawling's "I Made a Lover's Prayer" are sublime takes on criminally under heralded songs.

"I wasn't trying to find the most unheard of songs," Rodriguez says, calling from her Austin, Texas, home. "I just wanted to find songs that moved me. What's great about doing songs like 'Big Love' is that the only people who know of the song were the people that were really into Little Village. I like working with songs like that because you have more liberty with them. It's easier to bend the songs so much until the songs are your own. I can lie to myself and say, 'That's my song.'"

"I had so much fun with the more unknown songs but then again, I may have had the best time recording 'I'm So Lonesome I Could Cry.'"

Rodriguez flew to London to record that track because she was compelled to record with accomplished jazz guitarist Bill Frisell. Together, they created a surprising, stripped-down version of the classic.

"I was high on no sleep when I arrived in London to record with Bill," Rodriguez says. "I wanted to do something jazzy and I had to have Bill for what I wanted to do. It was so worth the effort to record the song with him."

Rodriguez will offer "I'm So Lonesome I Could Cry" and other cuts Wednesday at UNC Memorial Hall.

"I like doing the solo shows," she says. "It's fun to play these songs by myself. I'll also play some songs with [cellist] Ben Sollee, [who will open the show] but I'll primarily be out there on my own."

Rodriguez will also play songs from her first two albums, 2003's "The Trouble with Humans" and 2006's "Seven Angels on a Bicycle," both discs that failed to receive broad attention. The releases are filled with spirited, diverse originals that touch on country, blues and rock.

The beguiling Rodriguez, who recently moved back to Austin after a decade in Manhattan, will start writing new songs next year.

"I need to write after I'm finished touring," Rodriguez says. "If I write on the road, then I write about being on the road. I feel like that the only people who can relate to what I write on the road are musicians. So I'm going to write the next album when I'm done with the road.

"The great thing about 'Love and Circumstance' is that I feel like it cleared my mind when it comes to the next project. I'm ready to write and move on to the next thing, but I'll always look back fondly at this album. Doing a covers album was a great experience."